Newman's Tauberian Theorem

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Statement

Let $f:(0,+\infty)\to\mathbb C$ be a bounded function. Assume that its Laplace transform $F(s)=\int_0^\infty f(t)e^{-st}\,dt$ (which is well-defined by this formula for $\Re s>0$) admits an analytic extension (which we'll denote by the same letter $F$) to some open domain $E$ containing the closed half-plane $\{s\in\mathbb C\,:\,\Re s\ge 0\}$. Then the integral $\int_0^\infty f(t)\,dt$ converges and its value equals $F(0)$.

Proof

For every $T>0$, let $F_T(s)=\int_0^T f(t)e^{-st}\,dt$. The function $F_T$ is defined and analytic on the entire complex plane $\mathbb C$. The conclusion of the theorem is equivalent to the assertion $\lim_{T\to+\infty} F_T(0)=F(0)$. Now, choose some big $R>0$ and consider the contour $\Gamma=\Gamma_+\cup\Gamma_-$ as on the picture below.

Contour picture

Here $\Gamma_+$ is a semicircle and $\Gamma_-$ is any smooth curve that lies to the left of the imaginary axis except for its endpoints and such that the domain $D$ bounded by $\Gamma$ is entirely contained in $E$. By the Cauchy integral formula, we have

$F(0)-F_T(0)=\frac 1{2\pi i}\int_\Gamma K(z)(F(z)-F_T(z))\,dz$

where $K(z)$ is any kernel that is analytic in some neighbourhood of $D$ except for the point $0$ where it must have a simple pole with residue $1$.

The trick is to choose an appropriate kernel (depending on $T$) that makes the integral easy to estimate. To make a good choice, let us first estimate the difference $F(z)-F_T(z)$ on $\Gamma_+$. We have

$|F(z)-F_T(z)|= \left|\int_T^{+\infty}f(t)e^{-zt}\,dt\right| \le M\int_T^{+\infty}e^{-\Re z\,t}\,dt=M\frac {e^{-\Re z\,T}}{\Re z}$

where $M$ is a bound for $|f|$ on $(0,+\infty)$. Thus, we should kill the denominator $\Re z$ if we want the integral to converge. On the other hand, we can afford the kernel $K(z)$ grow as $e^{T\Re z}$ in the right half-plane (actually, we do not need any growth of $K(z)$ in the right half-plane for its own sake, but we need some decay in the left half-plane to estimate the integral over $\Gamma_-$ and it is impossible to get the latter without the first). This leads us to the choice

$K(z)=\left(\frac 1z+\frac z{R^2}\right)e^{Tz}$

Note that $K(\pm iR)=0$, so the unpleasant denominator $\Re z$ is, indeed, killed by $K$ on $\Gamma_+$. Also, $K$ decays in the left half-plane as fast as only is allowed by the exponential growth restriction in the right half-plane. This is not the only possible choice that will work, of course, but it is the simplest and the most elegant one.

Once this tricky choice is made, the rest is fairly straightforward. The integral over $\Gamma_+$ does not exceed

To be continued

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